Question

Solve the following equations which have equal roots:
$x^4-(a+b)x^3-a(a-b)x^2+a^2(a+b)x-a^{3}b=0$.

Answer 1

Give equation, $x^4-(a+b)x^3-a(a-b)x^2+a^2(a+b)x-a^{3}b=0$

Consider $f\left(x\right)=x^4-(a+b)x^3-a(a-b)x^2+a^2(a+b)x-a^{3}b$

$\therefore f^{\prime }\left(x\right)=4x^3-3(a+b)x^2-2a(a-b)x+a^2(a+b)$

Now, HCF of $f\left(x\right)$ and $f^{\prime }\left(x\right)$ is $(x-a)$. Hence $x=a$ is a double root of $f\left(x\right)=0$

$f\left(x\right)$ can be factored as $(x-a{)}^2(x^2+(a-b)x-ab)$

$\therefore$ roots of the given equation are $a,a,b,-a$